Effect of aligned magnetic field on the steady viscous flow past a circular cylinder

The steady viscous incompressible and slightly conducting fluid flow around a circular cylinder with an aligned magnetic field is simulated for the range of Reynolds numbers 100 ? Re ? 500 using the Hartmann number, M. The multigrid method with defect correction technique is used to achieve the second order accurate solution of complete non-linear Navier–Stokes equations. The magnetic Reynolds number is assumed to be small. It is observed that volume of the separation bubble decreases and drag coefficient increases as M is increased. We noticed that the upstream base pressure increases slightly with increase of M whereas downstream base pressure decreases with increase of M. The effect of the magnetic field on the flow is discussed with contours of streamlines, vorticity, plots of surface pressure and surface vorticity.     

Exact and approximation algorithms for the min–max k-traveling salesmen problem on a tree

Given k identical salesmen, where k ? 2 is a constant independent of the input size, the min–max k-traveling salesmen problem on a tree is to determine a set of k tours for the salesmen to serve all customers that are located on a tree-shaped network, so that each tour starts from and returns to the root of the tree with the maximum total edge weight of the tours minimized. The problem is known to be NP-hard even when k = 2. In this paper, we have developed a pseudo-polynomial time exact algorithm for this problem with any constant k ? 2, closing a question that has remained open for a decade. Along with this, we have further developed a (1 + ?)-approximation algorithm for any ? > 0.     

A finite difference solution to a two-dimensional parabolic inverse problem

In this paper, we consider the problem of finding u = u(x, y, t) and p = p(t) which satisfy u_t = u_xx + u_yy + p(t)u + ? in R × [0, T], u(x, y, 0) = f(x, y), (x, y) ∈ R = [0, 1] × [0, 1], u is known on the boundary of R and u(x^∗, y^∗, t) = E(t), 0 < t ? T, where E(t) is known and (x^∗, y^∗) is a given point of R. Through a function transformation, the nonlinear two-dimensional diffusion problem is transformed into a linear problem, and a backward Euler scheme is constructed. It is proved by the maximum principle that the scheme is uniquely solvable, unconditionally stable and convergent in L_∞ norm. The convergence orders of u and p are of O(τ + h²). The impact of initial data errors on the numerical solution is also considered. Numerical experiments are presented to illustrate the validity of the theoretical results.     

Convergence rate of the solution toward boundary layer solution for initial-boundary value problem of the 2-D viscous conservation laws

In this paper, we study the large time behavior of the solution to the initial boundary value problem for 2-D viscous conservation laws in the space x ? bt. The global existence and the asymptotic stability of a stationary solution are proved by Kawashima et al. [1]. Here, we investigate the convergence rate of solution toward the boundary layer solution with the non-degenerate case where f′(u₊) − b < 0. Based on the estimate in the H² Sobolev space and via the weighted energy method, we draw the conclusion that the solution converges to the corresponding boundary layer solution with algebraic or exponential rate in time, under the assumption that the initial perturbation decays with algebraic or exponential in the spatial direction.     

On single and double Soules matrices

Until now the concept of a Soules basis matrix of sign patternN consisted of an orthogonal matrix R∈R^n,n, generated in a certain way from a positive n-vector, which has the property that for any diagonal matrix Λ = diag(λ₁, … , λ_n), with λ₁ ? ? ? λ_n ? 0, the symmetric matrix A = RΛR^T has nonnegative entries only. In the present paper we introduce the notion of a pair of double Soules basis matrices of sign patternN which is a pair of matrices (P, Q), each in R^n,n, which are not necessarily orthogonal and which are generated in a certain way from two positive vectors, but such that PQ^T = I and such that for any of the aforementioned diagonal matrices Λ, the matrix A = PΛQ^T (also) has nonnegative entries only. We investigate the interesting properties which such matrices A have.As a preamble to the above investigation we show that the iterates, , generated in the course of the QR-algorithm when it is applied to A = RΛR^T, where R is a Soules basis matrix of sign pattern N, are again symmetric matrices generated by the Soules basis matrices R_k of sign pattern N which are themselves modified as the algorithm progresses.Our work here extends earlier works by Soules and Elsner et al.     

Generalized Gaussian quadrature rules over two-dimensional regions with linear sides

This paper presents a generalized Gaussian quadrature method for numerical integration over triangular, parallelogram and quadrilateral elements with linear sides. In order to derive the quadrature rule, a general transformation of the regions, R₁ = {(x, y)∣a ? x ? b, g(x) ? y ? h(x)} and R₂ = {(x, y)∣a ? y ? b, g(y) ? x ? h(y)}, where g(x), h(x), g(y) and h(y) are linear functions, is given from (x, y) space to a square in (ξ, η) space, S: {(ξ, η)∣0 ? ξ ? 1, 0 ? η ? 1}. Generlized Gaussian quadrature nodes and weights introduced by Ma et.al. in 1997 are used in the product formula presented in this paper to evaluate the integral over S, as it is proved to give more accurate results than the classical Gauss Legendre nodes and weights. The method can be used to integrate a wide class of functions including smooth functions and functions with end-point singularities, over any two-dimensional region, bounded by linear sides. The performance of the method is illustrated for different functions over different two-dimensional regions with numerical examples.     

Finite projective planes admitting a projective linear group PSL (2, q)

Let S be a projective plane, and let G?Aut(S) and PSL(2, q) ? G ? PΓL(2, q) with q > 3. If G acts point-transitively on S, then q = 7 and S is of order 2.     

Characterization of the trace by monotonicity inequalities

Let ?be a positive linear functional on the algebra of n × n complex matrices and p be a number greater than 1. The main result of the paper says that if for any pair A, B of positive semi-definite n × n matrices with A ? B the inequality ?(A^p) ? ?(B^p) holds true, then ?is a nonnegative scalar multiple of the trace.     

Batch delivery scheduling with batch delivery cost on a single machine

We consider a scheduling problem in which n independent and simultaneously available jobs are to be processed on a single machine. The jobs are delivered in batches and the delivery date of a batch equals the completion time of the last job in the batch. The delivery cost depends on the number of deliveries. The objective is to minimize the sum of the total weighted flow time and delivery cost. We first show that the problem is strongly NP-hard. Then we show that, if the number of batches is B, the problem remains strongly NP-hard when B ? U for a variable U ? 2 or B ? U for any constant U ? 2. For the case of B ? U, we present a dynamic programming algorithm that runs in pseudo-polynomial time for any constant U ? 2. Furthermore, optimal algorithms are provided for two special cases: (i) jobs have a linear precedence constraint, and (ii) jobs satisfy the agreeable ratio assumption, which is valid, for example, when all the weights or all the processing times are equal.     

A multi-round generalization of the traveling tournament problem and its application to Japanese baseball

In a double round-robin tournament involving n teams, every team plays 2(n − 1) games, with one home game and one away game against each of the other n − 1 teams. Given a symmetric n by n matrix representing the distances between each pair of home cities, the traveling tournament problem (TTP) seeks to construct an optimal schedule that minimizes the sum total of distances traveled by the n teams as they move from city to city, subject to several natural constraints to ensure balance and fairness. In the TTP, the number of rounds is set at r = 2. In this paper, we generalize the TTP to multiple rounds (r = 2k, for any k ? 1) and present an algorithm that converts the problem to finding the shortest path in a directed graph, enabling us to apply Dijkstra’s Algorithm to generate the optimal multi-round schedule. We apply our shortest-path algorithm to optimize the league schedules for Nippon Professional Baseball (NPB) in Japan, where two leagues of n = 6 teams play 40 sets of three intra-league games over r = 8 rounds. Our optimal schedules for the Pacific and Central Leagues achieve a 25% reduction in total traveling distance compared to the 2010 NPB schedule, implying the potential for considerable savings in terms of time, money, and greenhouse gas emissions.     

The impact of smooth W-grids in the numerical solution of singular perturbation two-point boundary value problems

This paper develops a semi-analytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed two-point boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the W-grid, which depends on the perturbation parameter ? ? 1. For problems on [0, 1] with a boundary layer at one end point, the local mesh width h_i = x_i+1 − x_i, with 0 = x₀ < x₁ < ? < x_N = 1, is condensed at either 0 or 1. Two simple 2nd order finite element and finite difference methods are combined with the new mesh, and computational experiments demonstrate the advantages of the smooth W-grid compared to the well-known piecewise uniform Shishkin mesh. For small ?, neither the finite difference method nor the finite element method produces satisfactory results on the Shishkin mesh. By contrast, accuracy is vastly improved on the W-grid, which typically produces the nominal 2nd order behavior in L², for large as well as small values of N, and over a wide range of values of ?. We conclude that the smoothness of the mesh is of crucial importance to accuracy, efficiency and robustness.     

Multi-degree reduction of tensor product Bézier surfaces with general boundary constraints

We propose an efficient approach to the problem of multi-degree reduction of rectangular Bézier patches, with prescribed boundary control points. We observe that the solution can be given in terms of constrained bivariate dual Bernstein polynomials. The complexity of the method is O(mn₁n₂) with m ? min(m₁, m₂), where (n₁, n₂) and (m₁, m₂) is the degree of the input and output Bézier surface, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined rectangular Bézier surfaces, the result is a composite surface of global C^r continuity with a prescribed r ? 0. In the detailed discussion, we restrict ourselves to r ∈ {0, 1}, which is the most important case in practical application. Some illustrative examples are given.     

Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation

The work presents a mathematical model describing the time fractional anomalous-diffusion process of a generalized Stefan problem which is a limit case of a shoreline problem. In this model, the governing equations include a fractional time derivative of order 0 < α ? 1 and variable latent heat. The approximate solution of the problem is obtained by homotopy perturbation method. The results thus obtained are compared graphically with the exact solutions. A brief sensitivity study is also performed.     

Bounding the gap between extremal Laplacian eigenvalues of graphs

Let G be a graph whose Laplacian eigenvalues are 0 = λ₁ ? λ₂ ? ? ? λ_n. We investigate the gap (expressed either as a difference or as a ratio) between the extremal non-trivial Laplacian eigenvalues of a connected graph (that is λ_n and λ₂). This gap is closely related to the average density of cuts in a graph. We focus here on the problem of bounding the gap from below.     

Fault-tolerant edge and vertex pancyclicity in alternating group graphs

In [J.-M. Chang, J.-S. Yang. Fault-tolerant cycle-embedding in alternating group graphs, Appl. Math. Comput. 197 (2008) 760-767] the authors claim that every alternating group graph AG_n is (n − 4)-fault-tolerant edge 4-pancyclic. Which means that if the number of faults ∣F∣ ? n − 4, then every edge in AG_n − F is contained in a cycle of length ?, for every 4 ? ? ? n!/2 − ∣F∣. They also claim that AG_n is (n − 3)-fault-tolerant vertex pancyclic. Which means that if ∣F∣ ? n − 3, then every vertex in AG_n − F is contained in a cycle of length ?, for every 3 ? ? ? n!/2 − ∣F∣. Their proofs are not complete. They left a few important things unexplained. In this paper we fulfill these gaps and present another proofs that AG_n is (n − 4)-fault-tolerant edge 4-pancyclic and (n − 3)-fault-tolerant vertex pancyclic.     

Bipanconnectivity of faulty hypercubes with minimum degree

In this paper, we consider the conditionally faulty hypercube Q_n with n ? 2 where each vertex of Q_n is incident with at least m fault-free edges, 2 ? m ? n − 1. We shall generalize the limitation m ? 2 in all previous results of edge-bipancyclicity. We also propose a new edge-fault-tolerant bipanconnectivity called k-edge-fault-tolerant bipanconnectivity. A bipartite graph is k-edge-fault-tolerant bipanconnected if G − F remains bipanconnected for any F ⊂ E(G) with ∣F∣ ? k. For every integer m, under the same hypothesis, we show that Q_n is (n − 2)-edge-fault-tolerant edge-bipancyclic and bipanconnected, and the results are optimal with respect to the number of edge faults tolerated. This not only improves some known results on edge-bipancyclicity and bipanconnectivity of hypercubes, but also simplifies the proof.     

Inequalities between ∥f(A + B)∥ and ∥f(A) + f(B)∥

The conjecture posed by Aujla and Silva [J.S. Aujla, F.C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003) 217-233] is proved. It is shown that for any m-tuple of positive-semidefinite n × n complex matrices A_j and for any non-negative convex function f on [0, ∞) with f(0) = 0 the inequality ?f(A₁) + f(A₂) + ? + f(A_m)? ? ? f(A₁ + A₂ + ? + A_m)? holds for any unitarily invariant norm ? · ?. It is also proved that ?f(A₁) + f(A₂) + ? + f(A_m)? ? f(?A₁ + A₂ + ? + A_m?), where f is a non-negative concave function on [0, ∞) and ? · ? is normalized.     

A reduction approach to the repeated assignment problem

We consider the repeated assignment problem (RAP), which is a K-fold repetition of the n × n linear assignment problem (LAP), with the additional requirement that no assignment can be repeated more than once. In actual applications K is typically much smaller than n. First, we derive upper and lower bounds respectively by a heuristic together with local search, and an efficient method solving the continuous relaxation. The latter also solves a Lagrangian relaxation, such that the related pegging test, to fix variables at zero or one, decomposes into K independent pegging tests to LAPs. These can be solved exactly by transforming them into all-pairs shortest path problems. Together with these procedures, we also employ a virtual pegging test and reduce RAP in size. Numerical experiments show that the reduced instances, with K ? n, can be solved exactly using standard MIP solvers.     

Analyzing the longitudinal effect of hypersonic flow past a conical cone via the perturbation method

To analyze the hypersonic flow past a conical cone, the variations of gasdynamic properties subjected to the longitudinal curvature effect by using the perturbation method. An outer perturbation expansion has been carried out by recent researchers, but a problem occurred, the outer expansion solutions are not uniformly valid in the shock layer, however, the outcome near the conical body surface called vortical layer remains deflective. This study intends to discover uniformly valid analytical solutions in the shock layer by applying the inner perturbation expansions matching with the out expansions to analyze the characteristics in the whole region including shock layer and vortical layer. Starting from the zero-order approximate solutions for hypersonic conical flow is then applied as the basic solutions for the outer perturbation expansions of a flow field. The governing equations and boundary conditions are also expanded via outer perturbations. Using an approximate analytical scheme in the derivation process, first-order perturbation equations can be simplified and the approximate closed-form solutions are obtained; furthermore, the various flow field quantities, including the normal force coefficient on the cone surface, have been calculated. According to the variations of gasdynamic properties, the longitudinal curvature effect for the hypersonic flow past a conical cone can be determined. Thicknesses of shock layer and vortical layer can be predicted as well. The physical phenomena inside both layers can be investigated carefully, the conditions for an elliptic cone with longitudinal curvature, m = 1 and n = 2 and other conditions of parameters; the perturbation parameter, ε_m2 = 0.1, semi-vertex angle of the unperturbed cone, δ = 10°, and hypersonic similarity parameter, K_δ = M_∞δ = 1.0, the thickness of vortical layer, η_VL, can be calculated at the position angle of conical cone body, ? = 30° was demonstrated here. Results show how very thin the vortical layer is approximately only 10% of the shock layer close to the body, the pressure in the whole shock layer is verified to be uniformly valid which agrees with previous studies. Large gradient changes in entropy and density are found when the flow approaches the cone surface, the most important is, this method provides a benchmark solution to the hypersonic flow past a conical cone and to assist the grids and numerics for numerical computation should be fashioned to accommodate the whole flow field region including the vortical layer of rapid adjustment, and let the analysis become more effective and low cost.